# Thread: difficult sin x function

1. ## difficult sin x function

Solve for x: (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27

2. Originally Posted by pacman
Solve for x: $\displaystyle (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x} = 26/27$
Something wrong here? If you put $\displaystyle y = \sin^2x$ then $\displaystyle 0\leqslant y\leqslant1$. But the function $\displaystyle (16/81)^y + (8/9)^{1-y}$ is greater than 1 throughout the interval [0,1], so it can never be equal to 26/27.

3. ok say instead of 26/27 it was 3/2.

how can you solve it for an exact soln? w/o using any approximating methods

4. Originally Posted by Krahl
ok say instead of 26/27 it was 3/2.

how can you solve it for an exact soln? w/o using any approximating methods

If It is $\displaystyle ( \frac{4}{9})^{\cos^2{x}}$ not

$\displaystyle ( \frac{8}{9})^{\cos^2{x}}$

Sub $\displaystyle (\frac{4}{9})^{\sin^2{x}} = u$then

$\displaystyle u^2 + (4/9) u^{-1} = M$

$\displaystyle u^3 - M u + (4/9) = 0$

It becomes a cubic equation

after solving u $\displaystyle u = (\frac{4}{9})^{\sin^2{x}}$

$\displaystyle x = \arcsin{ \sqrt{ \frac{ \ln{u} }{ \ln(4/9) } } }$

5. i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?

6. and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?

7. Originally Posted by pacman
i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?
My comment: it is not true that this is the plot of $\displaystyle y = (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x}$. Rather, this is the plot of $\displaystyle y = (16/81)^{sin^2 x} + (8/9)^{cos^2 x}-26/27$.

Also: This function assumes a minimum value of $\displaystyle \tfrac{19}{18}\approx 0.235$ at $\displaystyle x=\tfrac{\pi}{2}+n\cdot\pi, n\in\mathbb{Z}$. But it is always $\displaystyle >0$.

8. Originally Posted by pacman
and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?
It means that this time you have plotted $\displaystyle y=(16/81)^{sin^2 x} + (8/9)^{cos^2 x}$ for a larger range of x and, for some reason that I cannot fathom, seem to have surprised yourself with the result.

9. your observation is quite alright with me, thanks